空间L^p(Γ,Σ,μ)(1<p<∞)和Banach空间E的单位球面之间等距算子的延拓

Extension of Isometries Between the Unit Spheres of Lp((Γ,Σ,μ)(1<p<∞)and a Banach Space E

文章得到以下结果(它改进了文献[16][18]中的一些结果):设E是一个赋范空间,V0是单位球面S(Lp(Γ,Σ,μ))到单位球面S(E)内的等距映射。如果V0满足下列两个条件:(ⅰ)对于任意的自然数n,实数ξk∈[-1,1]及χAk∈χ(Γ),1≤k≤n,有‖sum from k=1 to n ξkμ(Ai)1/pV0〔(χAi)/(μ(Ai)1/p)〕‖p=sum from k=1 to n|ξk|pμ(Ai),(ⅱ)对于任意的f1,f2∈S(Lp(Γ,Σ,μ))和实数ξ1,ξ2∈[-1,1],有‖ξ1V0(f1)+ξ2V0(f2)‖=1|ξ1V0(f1)+ξ2V0(f2)∈V0[S(Lp(Γ,Σ,μ)],那么V0可延拓为全空间Lp(Γ,Σ,μ)上的等距线性算子。

In this paper,we show the following result which generalizes some theorems（see）：Let E be a normed space,V0：S（Lp（（Γ,,μ））→S（E）be a isometric mapping.If V0 satisfies the following two hypotheses：（ⅰ）For all n∈and χAk∈χ（Γ）,1≤k≤n,such that ‖sum from k=1 to n ξkμ（Ai）1/pV0〔（χAi）/（μ（Ai）1/p）〕‖p=sum from k=1 to n｜ξk｜pμ（Ai）,（ⅱ）For every f1,f2∈S（Lp（Γ,Σ,μ））and ξ1,ξ2∈,such that ‖ξ1V0（f1）＋ξ2V0（f2）‖=1｜ξ1V0（f1）＋ξ2V0（f2）∈V0[S（Lp（Γ,Σ,μ）], then V0can be extended to a linear isometry defined on the whole space Lp（Γ,Σ,μ）.